Average Error: 0.5 → 0.5
Time: 13.1s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \left(\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right) + \left(\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\left(-\frac{\sqrt[3]{\sin x}}{16}\right) + \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{2 + \left(\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right) + \left(\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\left(-\frac{\sqrt[3]{\sin x}}{16}\right) + \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
double f(double x, double y) {
        double r174086 = 2.0;
        double r174087 = sqrt(r174086);
        double r174088 = x;
        double r174089 = sin(r174088);
        double r174090 = y;
        double r174091 = sin(r174090);
        double r174092 = 16.0;
        double r174093 = r174091 / r174092;
        double r174094 = r174089 - r174093;
        double r174095 = r174087 * r174094;
        double r174096 = r174089 / r174092;
        double r174097 = r174091 - r174096;
        double r174098 = r174095 * r174097;
        double r174099 = cos(r174088);
        double r174100 = cos(r174090);
        double r174101 = r174099 - r174100;
        double r174102 = r174098 * r174101;
        double r174103 = r174086 + r174102;
        double r174104 = 3.0;
        double r174105 = 1.0;
        double r174106 = 5.0;
        double r174107 = sqrt(r174106);
        double r174108 = r174107 - r174105;
        double r174109 = r174108 / r174086;
        double r174110 = r174109 * r174099;
        double r174111 = r174105 + r174110;
        double r174112 = r174104 - r174107;
        double r174113 = r174112 / r174086;
        double r174114 = r174113 * r174100;
        double r174115 = r174111 + r174114;
        double r174116 = r174104 * r174115;
        double r174117 = r174103 / r174116;
        return r174117;
}
↓
double f(double x, double y) {
        double r174118 = 2.0;
        double r174119 = y;
        double r174120 = sin(r174119);
        double r174121 = cbrt(r174120);
        double r174122 = r174121 * r174121;
        double r174123 = x;
        double r174124 = sin(r174123);
        double r174125 = cbrt(r174124);
        double r174126 = 16.0;
        double r174127 = r174125 / r174126;
        double r174128 = r174125 * r174125;
        double r174129 = -r174128;
        double r174130 = r174127 * r174129;
        double r174131 = fma(r174122, r174121, r174130);
        double r174132 = sqrt(r174118);
        double r174133 = r174131 * r174132;
        double r174134 = r174120 / r174126;
        double r174135 = r174124 - r174134;
        double r174136 = r174133 * r174135;
        double r174137 = -r174127;
        double r174138 = r174137 + r174127;
        double r174139 = r174128 * r174138;
        double r174140 = r174139 * r174132;
        double r174141 = r174140 * r174135;
        double r174142 = r174136 + r174141;
        double r174143 = cos(r174123);
        double r174144 = cos(r174119);
        double r174145 = r174143 - r174144;
        double r174146 = r174142 * r174145;
        double r174147 = r174118 + r174146;
        double r174148 = 3.0;
        double r174149 = 1.0;
        double r174150 = 5.0;
        double r174151 = sqrt(r174150);
        double r174152 = r174151 - r174149;
        double r174153 = r174152 / r174118;
        double r174154 = r174153 * r174143;
        double r174155 = r174149 + r174154;
        double r174156 = r174148 - r174151;
        double r174157 = r174156 / r174118;
        double r174158 = r174157 * r174144;
        double r174159 = r174155 + r174158;
        double r174160 = r174148 * r174159;
        double r174161 = r174147 / r174160;
        return r174161;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.5

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{\color{blue}{1 \cdot 16}}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  4. Applied add-cube-cbrt0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}}}{1 \cdot 16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  5. Applied times-frac0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \color{blue}{\frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1} \cdot \frac{\sqrt[3]{\sin x}}{16}}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  6. Applied add-cube-cbrt0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}\right) \cdot \sqrt[3]{\sin y}} - \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1} \cdot \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  7. Applied prod-diff0.5

    \[\leadsto \frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, -\frac{\sqrt[3]{\sin x}}{16} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{\sin x}}{16}, \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}, \frac{\sqrt[3]{\sin x}}{16} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}\right)\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  8. Applied distribute-lft-in0.5

    \[\leadsto \frac{2 + \color{blue}{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, -\frac{\sqrt[3]{\sin x}}{16} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}\right) + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\sin x}}{16}, \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}, \frac{\sqrt[3]{\sin x}}{16} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}\right)\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  9. Simplified0.5

    \[\leadsto \frac{2 + \left(\color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)} + \left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{\sin x}}{16}, \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}, \frac{\sqrt[3]{\sin x}}{16} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{1}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  10. Simplified0.5

    \[\leadsto \frac{2 + \left(\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right) + \color{blue}{\left(\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\left(-\frac{\sqrt[3]{\sin x}}{16}\right) + \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  11. Final simplification0.5

    \[\leadsto \frac{2 + \left(\left(\mathsf{fma}\left(\sqrt[3]{\sin y} \cdot \sqrt[3]{\sin y}, \sqrt[3]{\sin y}, \frac{\sqrt[3]{\sin x}}{16} \cdot \left(-\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right) + \left(\left(\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \left(\left(-\frac{\sqrt[3]{\sin x}}{16}\right) + \frac{\sqrt[3]{\sin x}}{16}\right)\right) \cdot \sqrt{2}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))